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form_factor

form_factor#

import ampform.dynamics.form_factor

Implementations of the form factor, or barrier factor.

class FormFactor(s, m1, m2, angular_momentum, meson_radius, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

Formulate a Blatt-Weisskopf form factor.

Returns the production process factor \(n_a\) from Equation (50.26) in PDG2021, §Resonances, p.9, which features the sqrt of a BlattWeisskopfSquared.

(1)#\[\begin{split} \begin{array}{rcl} \mathcal{F}_{L}\left(s, m_{a}, m_{b}\right) &=& \sqrt{B_{L}^2\left(d^{2} q^2\left(s\right)\right)} \\ \end{array}\end{split}\]
class BlattWeisskopfSquared(z, angular_momentum, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

Normalized Blatt-Weisskopf function \(B_L^2(z)\), with \(B_L^2(1)=1\).

Parameters:

  • z – Argument of the Blatt-Weisskopf function \(B_L^2(z)\). A usual choice is \(z = (d q)^2\) with \(d\) the impact parameter and \(q\) the breakup-momentum (see BreakupMomentumSquared).

  • angular_momentum – Angular momentum \(L\) of the decaying particle.

Note that equal powers of \(z\) appear in the nominator and the denominator, while some sources define an non-normalized form factor \(F_L\) with \(1\) in the nominator, instead of \(z^L\). See for instance Equation (50.27) in PDG2021, §Resonances, p.9. We normalize the form factor such that \(B_L^2(1)=1\) and that \(B_L^2\) is unitless no matter what \(z\) is.

See also

Form factor, TR-029, and [Chung, 2015].

With this, the implementation becomes

(2)#\[\begin{split} \begin{array}{rcl} B_{L}^2\left(z\right) &=& \frac{\left|{h_{L}^{(1)}\left(1\right)}\right|^{2}}{z \left|{h_{L}^{(1)}\left(\sqrt{z}\right)}\right|^{2}} \\ \end{array}\end{split}\]

where \(h_{L}^{(1)}\left(z\right)\) is defined by (3).

class SphericalHankel1(l, z, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

Spherical Hankel function of the first kind for real-valued \(z\).

See [von Hippel and Quigg, 1972], Equation (A12), and TR-029 for more info. This page explains the difference with the general Hankel function of the first kind, \(H_\ell^{(1)}\).

This expression class assumes that \(z\) is real and evaluates to the following series:

(3)#\[\begin{split} \begin{array}{rcl} h_{\ell}^{(1)}\left(z\right) &=& \frac{\left(- i\right)^{\ell + 1} e^{i z} \sum_{k=0}^{\ell} \frac{\left(\frac{i}{2 z}\right)^{k} \left(k + \ell\right)!}{k! \left(- k + \ell\right)!}}{z} \\ \end{array}\end{split}\]
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